Page 338 - Applied Numerical Methods Using MATLAB

UNCONSTRAINED OPTIMIZATION 327

function [xo,fo] = Nelder0(f,abc,fabc,TolX,TolFun,k)
[fabc,I] = sort(fabc); a = abc(I(1),:); b = abc(I(2),:); c = abc(I(3),:);
fa = fabc(1); fb = fabc(2); fc = fabc(3); fba = fb - fa; fcb = fc - fb;
ifk<=0 | abs(fba) + abs(fcb) < TolFun | abs(b - a) + abs(c - b) < TolX
xo=a;fo=fa;
if k == 0, fprintf(’Just best in given # of iterations’), end
else
m = (a + b)/2; e = 3*m - 2*c; fe = feval(f,e);
if fe < fb, c = e; fc = fe;
else
r = (m+e)/2; fr = feval(f,r);
if fr < fc, c = r; fc = fr; end
if fr >= fb
s = (c + m)/2; fs = feval(f,s);
if fs < fc, c = s; fc = fs;
elseb= m;c=(a+ c)/2; fb = feval(f,b); fc = feval(f,c);
end
end
end
[xo,fo] = Nelder0(f,[a;b;c],[fa fb fc],TolX,TolFun,k - 1);
end
function [xo,fo] = opt_Nelder(f,x0,TolX,TolFun,MaxIter)
N = length(x0);
if N == 1 %for 1-dimensional case
[xo,fo] = opt_quad(f,x0,TolX,TolFun); return
end
S = eye(N);
for i = 1:N %repeat the procedure for each subplane
i1=i+1;if i1>N,i1=1;end
abc = [x0; x0 + S(i,:); x0 + S(i1,:)]; %each directional subplane
fabc = [feval(f,abc(1,:)); feval(f,abc(2,:)); feval(f,abc(3,:))];
[x0,fo] = Nelder0(f,abc,fabc,TolX,TolFun,MaxIter);
ifN<3, break; end %No repetition needed for a 2-dimensional case
end
xo = x0;
%nm713.m: do_Nelder
f713 = inline(’x(1)*(x(1)-4-x(2)) +x(2)*(x(2)-1)’,’x’);
x0 = [0 0], TolX = 1e-4; TolFun = 1e-9; MaxIter = 100;
[xon,fon] = opt_Nelder(f713,x0,TolX,TolFun,MaxIter)
%minimum point and its function value
[xos,fos] = fminsearch(f713,x0) %use the MATLAB built-in function

This program also applies the MATLAB built-in routine “fminsearch()”to min-
imize the same objective function for practice and confirmation. The minimization
process is illustrated in Fig. 7.4.

(cf) The MATLAB built-in routine “fminsearch()” uses the Nelder–Mead algorithm
to minimize a multivariable objective function. It corresponds to “fmins()”inthe
MATLAB of version.5.x.